Problem: Let $C$ be the curve defined by $x^2 - 3xy + y = 0$. We have a change of variables: $\begin{aligned} x &= X_1(u, v) = 3v + 2u \\ \\ y &= X_2(u, v) = u - 2v \end{aligned}$ What is $C$ under the change of variables? Choose 1 answer: Choose 1 answer: (Choice A) A $18v^2 + 3uv + 4u^2 + u - 2v = 0$ (Choice B) B $9v^2 + 24uv - 6u^2 + u - 2v = 0$ (Choice C) C $-6v^2 - 9uv + 2u^2 + u - 2v = 0$ (Choice D) D $27v^2 + 15uv - 2u^2 + u - 2v = 0$
Answer: When applying a change of variables, we substitute the new definition for $x$ and $y$ into the original equation. The original equation: $x^2 - 3xy + y = 0$ Let's substitute $X_1(u, v)$ for $x$ and $X_2(u, v)$ for $y$. $\begin{aligned} (3v + 2u)^2 - 3(3v + 2u)(u - 2v) + (u - 2v) &= 0 \\ \\ 9v^2 + 12uv + 4u^2 - 3(3uv - 6v^2 + 2u^2 - 4uv) + u - 2v &= 0 \\ \\ 9v^2 + 12uv + 4u^2 - 9uv + 18v^2 - 6u^2 + 12uv + u - 2v &= 0 \\ \\ 27v^2 + 15uv - 2u^2 + u - 2v &= 0 \end{aligned}$ Therefore, under the change of variables, $C$ becomes: $27v^2 + 15uv - 2u^2 + u - 2v = 0$